Forwards & Futures
Regular Forward
You agree now to buy an asset at time t and the money is exchanged at time t. This equals what we expect the stock to be worth at time t.
- \(F=S_0 e^{(r-\delta)t}\)
- Continuous Dividends
- Compound by r, discount by \(\delta\) b/c you aren’t receiving the dividends
- \(F=[S_0 - \sum (Div_i)(e^{-rt_i})]e^{rt}\)
- Discrete Dividends
- Discount each dividend to the PV so you get them all at the same time, then compound everything to time t. We use the rfr for all discounting and compounding here.
Prepaid Forward
You pay upfront to receive the asset at time t. Asset holder still receives dividends during this time.
- \(F^P = S_0e^{-\delta t}\)
- Continuous dividends
- Just discount the stockprice by the dividends b/c you don’t get these
- \(F^P = S_0 - \sum (Div_i)(e^{-rt_i})\)
- Discrete dividends are discounted by the continuous rfr rate
Symbols
- \(F\): Forward price you pay at time t
- \(F^P\): Prepaid forward price you pay at time 0
- \(S_0\) or just \(S\): Stock price at t=0
- \(r\): Risk free interest rate
- \(\delta\): continuously compounded dividend rate
- \(t\): time until expiration
Relationship between regular and prepaid forwards
- \(F = F^Pe^{rt}\)
- Regular forwards equations are just the prepaid forward equations compounded by \(e^{rt}\)
Currency Forwards
- Think of currency forwards just like asset forwards. But that you are receiving a foreign currency as the asset. The symbols and language is different but you can still use the above formulas.
- \(F = x_0e^{(r-r_f)t}\)
- \(F^P = x_0e^{-r_f(t)}\)
- \(x_0\) = Exchange Rate, how much you have to pay in base currency to receive the foreign currency. (\(S_0\))
- \(r\) = rfr for the base currency you currently have
- \(r_f\) = rfr for the foreign currency you are buying (\(\delta\))
Futures
- Futures use the same formulas as forwards
- Futures are traded on exchanges while forwards are traded over the counter
- Futures are cash settled, and deal with margins
- Some language
- Notional Value = Index * Size
- Margin = %age of notional value that you have to keep in your margin account (in case you default)
- Maintenance Margin = %age of your original margin balance that is the minimum allowed amount. You will get a marign call if you have less money than the maintenance margin and have to deposit more money.
- Ex: You enter into a futures contract to long the S&P 500. The size is $200, the index is 700, margin is 25%, maintenance margin is 80%, and risk free rate is .05.
- Balances at t=0:
- Notional Value = Index * Size = \(\$700(200) = \$140,000\)
- Initial balance in the margin account = (Notional Value)(margin %) = \(\$140,000(.25) = \$35,000\)
- Minimum balance before a margin call = \(\$35,000(.8) = \$28,000\)
- 1 week passes. Index price rises to 730.
- Margin account balance grows by rfr for 1 week: \(\$35,000e^{.05(\frac{1}{52})}=3.503367\times 10^{4}\)
- Index price rose by \(730-700=30\), so margin balance increases by \(30*200=6,000\)
- Margin balance at t = 1 week:
- \(=\$35,000e^{.05(\frac{1}{52})} + 6,000=4.103367\times 10^{4}\)
- Another week passes. t = 2 weeks. Index price falls to $650.
- Margin grows by rfr, and margin balance decreases by \(730-650=\$80\) times the size of 200
- Margin Balance at t = 2 weeks \(=\$4.103367\times 10^{4}e^{(\frac{.05}{52})} - (80)(200)=2.507314\times 10^{4}\)
- So we would get a margin call to add more money to the margin account since the value dropped below the maintenance margin of $28,000
- Balances at t=0:
Definitions
- Forward Premium = \(\frac{F}{S_0}\)
- Annualized Forward Premium = \(\frac{1}{t}Log(\frac{F}{S_0})=r-\delta\)
- (Expressed as a percentage)
- Risk Prmeium = \(\alpha - r\)
- Rate of Appreciation = \(\alpha - \delta\)
- AKA: Continuously compounded expected rate of appreciation
Options Basics
Overview
- Main types
- European = can only be exercised at expiration, easier math so assume this
- American = can be exercised at any point prior to expiration, must use binomial trees for pricing this
- Exotic Types
- Asian, Barrier, Compound, Gap, Exchange
- These are explained in later sections
- Payoffs
- Long Call = max{0, Spot - Strike}
- Buying the option to buy the stock at the strike price
- Hoping stock goes up, so you can buy at low strike, and sell at high spot
- Long Put = max{0, Strike - Spot}
- Buying the option to sell the stock at the strike price
- Hoping stock goes down, so you can buy the stock at the low spot, and sell at a high strike
- Short Call = min{0, Strike - Spot}
- Selling someone the option to buy the stock from you at the strike price
- Hoping the stock will go down so the option you sold is never exercised w/ 0 payoff. So you just profit the premium.
- Short Put = min{0, Spot - Strike}
- Selling someone the option to sell you the stock at the strike price
- Hoping the stock goes up, so the spot > strike, so they won’t sell it to you at the lower price. So you just profit the premium.
- Long Call = max{0, Spot - Strike}
- Profits
- = payoff + FV of the premium (compounded by the rfr)
- Language
- Spot price: actual price of the stock at expiration
- Strike: agreed upon price that will be exchanged for the asset
General strategy for dealing with solo call/put payoffs
- Look at the left and right tails
- If the LEFT side goes up/down => Put
- If the RIGHT side goes up/down => Call
- If it goes UP => Long
- If it goes DOWN => Put
Options Profit Plots
Option + Asset Combinations
Floor
Cap
Covered Call
Covered Put
Options Spreads
- Spreads mean it’s using all calls or all puts. But this definition is fudged a bit.
Bull Spread
- 2 Calls OR 2 Puts.
- Buy LOW & Sell HIGH
Bear Spread
- 2 Calls OR 2 Puts
Buy HIGH & Sell LOW
2 Calls OR 2 Puts
Box Spread
- Rare to see on IFM
- Box spreads are a 0 profit spread, so it’s the same as lending/borrowing money
- A 4 option strategy consisting of a Bull Spread + Bear Spread
- 2 ways to build, and this changes whether it’s equivalent to borrowing or lending money
- Bull Spread w/ Calls + Bear Spread w/ Puts => Lending money at rfr (buying bonds)
- Bull Spread w/ Puts + Bear Spread w/ Calls => Borrowing money at rfr (shorting bonds)
- All using only 2 strikes (\(K_1\) & \(K_2\))
Ratio Spread
- Combination of buying/selling m options at one strike price, and n options at a different strike price.
Collars
Purchased Collar
- BUY Put low, WRITE Call High.
- So it’s flat in the middle
- Collar Width: Distance between the Put and the Call
- Weird that a Purchased collar looks like a bearish position
Written Collar
- WRITE Put Low, LONG Call High
Zero Cost Collar
- A Collar Spread (either purchased or written) that you break even (0 profit) in the middle
Long Straddle
- Buy a call and a put at the same strike price
- Betting on high volatility
Written Straddle
- Writting both a call and a put at the same strike price
- Betting on low volatility
Strangle
- Long both a call and a put at different strike prices (it doesn’t matter which one is low/high)
Written Strangle
- Write both a call and a put at different strike prices
Butterfly Spread
Combination of 4 option positions at 3 different strike prices (\(K_1 < K_2 < K_3\))
For a purchased Butterfly Spread, there are several ways to construct.
- Main way to construct (using all calls or all puts):
- All Calls:
- Buy \(C(K_1)\)
- Sell 2 \(C(K_2)\)
- Buy \(C(K_3)\)
- All Puts:
- Buy \(P(K_1)\)
- Sell 2 \(P(K_2)\)
- Buy \(P(K_3)\)
- Another way to think of this is a combination of a Bull & Bear spread
- All Calls:
- Using a combination of calls and puts:
- Buy \(P(K_1)\)
- Sell \(P(K_2)\)
- Sell \(C(K_2)\)
- Buy \(C(K_3)\)
- Another way to think of this is to write a Straddle at \(K_2\) + Buy a Strangle w/ strikes \(K_1\) & \(K_3\)
- When constructed this way it is called an “Iron Butterfly”
Each of these positions create this identical plot:
- Written Butterfly
- Switch long/short positions from above
Asymmetric Butterfly
- Still work w/ 3 strike prices \(K_1 < K_2 < K_3\)
- But this time \(K_2\) isn’t exactly in the middle
Coaching Actuaries Way:
- Long (\(K_3-K_2\)) options at \(K_1\)
- Short (\(K_3-K_1\)) options at \(K_2\)
- Long (\(K_2-K_1\)) options at \(K_3\)
- How many you long/short at a certain strike price, is the difference between the other 2 strike prices
- Much simpler than the method below that comes from the textbook:
Textbook way:
To construct an asymetric butterfly w/ all calls:
You have some \(\lambda\) where:
- \(\lambda = \frac{K_3 - K_2}{K_3 - K_1}\)
- Solving for \(K_2 = \lambda K_1 + (1- \lambda)K_3\)
- So for every \(K_2\) Call you write, you must:
- Buy \(\lambda K_1\) Calls
- Buy \((1-\lambda) K_3\) Calls
Ex: Suppose you wish to create an asymetric butterfly with strikes 100, 110, and 115. You wish to buy/sell 12 options total. How many of each option do you buy/sell?
- \(\lambda = \frac{K_3 - K_2}{K_3 - K_1} = \frac{115 - 110}{115 - 100} = \frac{1}{3}\)
So our ratios are as follows:
- 1 \(K_2\) Call sold
- 1/3 \(K_1\) calls bought
- 2/3 \(K_3\) calls bought
Want 12 options total:
- \(1x + \frac{1}{3}x + \frac{2}{3}x = 12\)
- \(x = 6\)
Multiply the above ratios by 6 each to see how many of each to buy
Iron Condor
- Like a butterfly, but with space between the options at the point. So it has 4 strikes \(K_1 < K_2 < K_3 < K_4\)
- Not even sure if this is tested on IFM but it’s easy
Synthetic Forward
- Buying a call and Selling a put at the same strike price w/ the same expiration mimics a forward contract.
Arbitrage
Random things to know:
- American options should be worth more than European options, b/c they have the added flexibility to exercise at anytime.
- Never rational to exercise an American Call early on a non dividend paying stock
- B/c the only benefit to exercising early is to receive the high dividends, but you could be earning risk free interest in the mean time if there are no dividends.
- American Call Prem = European Call Prem, when there are no dividends
- When American options make sense to exercise early:
- American Calls make sense to exercise early when: PV(Dividends) > PV(Interest on Strike) + PV(Insurance)
- American Puts make sense to exercise early when: PV(Interest on Strike) > PV(Dividends) + PV(Insurance)
- PV(Insurance) means the premium
- You exercise early when you can recive more on the dividends or interest basically
3 cases when arbitrage exists
| Proposition | Calls | Puts |
|---|---|---|
| 1 | \(C(K_1) \geq C(K_2)\) | \(P(K_1) \leq P(K_2)\) |
| 2 | \(C_{Amer}(K_1)-C_{Amer}(K_2) \leq K_2 -K_1\) \(C_{Eur}(K_1)-C_{Eur}(K_2) \leq (K_2-K_1)e^{-rT}\) |
\(P_{Amer}(K_2)-P_{Amer}(K_1) \leq K_2 -K_1\) \(P_{Eur}(K_2)-P_{Eur}(K_1) \leq (K_2-K_1)e^{-rT}\) |
| 3 | \(\frac{C(K_1)-C(K_2)}{K_2-K_1} \geq \frac{C(K_2)-C(K_3)}{K_3-K_2}\) \(C(K_2)\leq \lambda C(K_1) + (1-\lambda)C(K_3)\) |
\(\frac{P(K_2)-P(K_1)}{K_2-K_1} \leq \frac{P(K_3)-P(K_2)}{K_3-K_2}\) \(P(K_2) \leq \lambda P(K_1) + (1-\lambda)P(K_3)\) |
In Words:
- Self explanatory, premiums rise when more in the money
- Premiums move slower than strikes. If you increase the strike by $5, the premium will change by less than $5.
- \(\Delta\)Premium < \(\Delta\)Strike
- A certain convexity exists. When more in the money, the ratio of difference of premiums to the difference of strikes will be greater (closer to 1). Remember than when you move way deep into the money, the profit becomes almost guaranteed, so the change in premium will almost equal the change in strike.
- Pearson textbook formula for this theorem. (This is easier to memorize).
- \(V(K_2) \leq \lambda V(K_1)+(1-\lambda)V(K_3)\)
- \(\lambda = \frac{K_3-K_2}{K_3-K_1}\)
- To exploit this, we short what’s more expensive, and long what’s cheaper.
- \(V(K_2) \leq \lambda V(K_1)+(1-\lambda)V(K_3)\)
- Pearson textbook formula for this theorem. (This is easier to memorize).
- To exploit the arbitrage in any of the above formulas:
- Move everything to the greater than side of the equation (formulas > 0).
- \(+\) values => Sell/Borrow (getting money)
- \(-\) values -> Buy/Lend (spending money)
- Ex: \(P(65)-P(60)-5>0\)
- We Sell P(65), Buy P(60), and Lend $5 (buy bond)
Option Pricing
Basic Profit Functions
- Profit Long Call = \(Max[0, S_T - K] - FV(Prem)\)
- Profit Short Call = \(-Max[0, S_T - K] + FV(Prem)\)
- Profit Long Put = \(Max[0, K - S_T] - FV(Prem)\)
Profit Short Put = \(-Max[0, K-S_T]+FV(Prem)\)
- Short & Long of each are just multiplied by -1
- Calls are \(S_T-K\), and Puts are \(K-S_T\)
Put Call Parity
This parity must exist between the Call and Put premium for the same asset with the same strikes and expiration dates, or else arbitrage exists.
For Stocks
- \(C - P = Se^{-\delta t} - Ke^{-rt}\)
- \(C - P + Ke^{-rt} = Se^{-\delta t}\)
- Can rearrange so it looks like this. The left side of the equation is a synthetic prepaid forward, the right side of the equation is the actual prepaid forward.
For currency
- \(C-P = x_0e^{-r_f t}-Ke^{-rt}\)
- Symbols
- C = Call Premium
- P = Put Premium
- S = \(S_0\) Stock price at t=0
- \(x_0\) = Foreign currency price at t=0
- \(\delta\) = dividend yield of stock
- \(r_f\) = rfr of foreign currency (\(\delta\))
- Just use 1st equation and treat foreign currency like the stock
Binomial Trees
1 Period Binomial Tree
Assume stock can either rise from \(S_0\) to \(S_u\), or go down to \(S_d\) at time t.
- Option Premium = \(\Delta S + B\)
- \(\Delta = e^{-\delta t} \frac{V_u - V_d}{S(u - d)}\)
- \(B=e^{-rt} \frac{u V_d - d V_u}{u-d}\)
- \(V_u/V_d\) = Option payoff if stock rises/drops. For call or put
- \(S\) = \(S_0\) = Stock price at t=0
- \(u\) & \(d\) = Ratio of increase or decrease
- \(u = \frac{S_1}{S_0}\)
- \(d = \frac{S_0}{S_1}\)
For calls we’ll get a \(+\Delta\) & \(-B\). For puts we’ll get a \(-\Delta\) and \(+B\)
- Synthetic position
- Can use this formula to create an equivalent synthetic position
- Call payoff = purchasing \(+\Delta\) shares of stock, and borrowing \(-B\) to do so
- Put payoff = shorting \(-\Delta\) shares of stock, and lending \(+B\)
Arbitrage
- If the Option Prem \(\neq\) \(\Delta S + B\), then we buy what’s lower, and short what’s higher
- Profit = \(|C^* - C|e^{rt}\)
- Difference between what premium actually is and what it should be
- Profit is calculated at time t=t
Risk Neutral Pricing with Binomial Trees
Risk Neutral Pricing means to assume the stock’s expected return is equal to the risk free rate
This time we incorporate probabily that the stock rises/decreases between now and the next time interval
\(Prem = e^{-rt}[p(C_u)+(1-p)C_d]\)
- \(p = \frac{e^{(r-\delta)t}-d}{u-d}\)
- \(u=e^{(r-\delta)t+\sigma \sqrt{t}}\)
- \(d=e^{(r-\delta)t-\sigma \sqrt{t}}\)
- Can switch \(C_u\) <==> \(P_u\)
- \(p = \frac{e^{(r-\delta)t}-d}{u-d}\)
p doens’t really equal the probability that the stock rises, but the risk-neutral probability, but you can just think of it this way in the formula
Volatility (\(\sigma\)) is calculated in 3 steps
- Find log ratios
- \(log(\frac{S_t}{S_{t-h}})\)
- Find unbiased estimator of the variance of these ratios
- \(S^2 = \frac{1}{n-1}\sum(x_i-\bar{x})^2\)
- Or just plug values into multiview
- Multiply by 1/h and square root to get SD
- \(\sigma = \sqrt{S^2/h}\)
- (For weekly values, h = 1/52, so we’d multiply it by 52)
- \(\sigma = \sqrt{S^2/h}\)
- Find log ratios
Price using multiple period tree
European Pricing
- Start with very right big side of the tree and find all payoffs there
- Then we sum up all of these payoff possibilities, times their probabilities (w combinatorics factor), and discount
- \(Prem = e^{-rt}[\sum \binom{m}{k} (prob_i)(payoff_i)]\)
- Combinatorics factor is the total number of periods (4 in this pic), choose either of the probability exponents
- \(p = \frac{e^{(r-\delta)h}-d}{u-d}\)
- \(u=e^{(r-\delta)h + \sigma \sqrt{h}}\)
- \(d=e^{(r-\delta)h - \sigma \sqrt{h}}\)
- We replace t w/ h which is our step size
American Pricing
- American -> can exercise option at any time
- Key is to work backwards
- Find all payoffs at end period
- Work backwards step by step
- For each step back you take the max between:
- Immediate payoff if exercised immediately
- \(e^{-rh}[pC_u + (1-p)C_d]\)
- For each step back you take the max between:
- Repeat until you work your way back to the first step
- Can also price European Options this way, but it just takes forever
Options on Futures Contracts
Treat very similar to stocks, but you substitute F for S and r for \(\delta\) in the equations. And the formulas for \(\Delta\) and \(B\) for the replicating portfolio formulas are slightly different.
- \(V_0=\Delta F + B\)
- \(\Delta = \frac{V_u-V_d}{F (u_F-d_F)}\)
- (Change in Options Prices divided by Change in Futures Prices)
- \(u_F = e^{(r-r)h+\sigma \sqrt{h}}=e^{\sigma \sqrt{h}}\)
- \(d_F = e^{(r-r)h- \sigma \sqrt{h}}=e^{-\sigma \sqrt{h}}\)
- \(B=e^{-rh}[pV_u+(1-p)V_d]\)
- \(p=\frac{e^{(r-r)h}-d_F}{u_F-d_F}=\frac{1-d_F}{u_F-d_F}\)
- Just remember the regular formulas for u, d, and p, and substitute r for \(\delta\) and F for S.
- \(p=\frac{e^{(r-r)h}-d_F}{u_F-d_F}=\frac{1-d_F}{u_F-d_F}\)
- \(\Delta = \frac{V_u-V_d}{F (u_F-d_F)}\)
Then I’m pretty sure you have to use the American Option method, where you use this at every single node, and work your way backwards down the multiple period tree
Options on Currencies
- Just treat the foreign currency like a stock
- Exchange rate = \(x_0\) => \(S_0\)
- Domestic currency rfr = \(r_d\) => \(r\)
- Foreign currency rfr = \(r_f\) => \(\delta\)
Black Scholes
- This is how modern options are priced today. Based on the lognormal distribution.
Risk Neutral Measure
- Assume \(\alpha = \gamma=r\)
- Expected return of stock and option are both equal to the rfr
- \(C = Se^{-\delta t}N(d_1) - Ke^{-rt}N(d_2)\)
- \(P = Ke^{-rt}N(-d_2) - Se^{-\delta t}N(-d_1)\)
- \(d_1 = \frac{log(\frac{S}{K}) + (r - \delta + .5 \sigma^2)t}{\sigma \sqrt{t}}\)
- \(d_2 = d_1 - \sigma \sqrt{t} = \frac{log(\frac{S}{K}) + (r - \delta - .5 \sigma^2)t}{\sigma \sqrt{t}}\)
- Just use 1st formula, but you can also use the second. Only difference is the minus \(.5\sigma^2\)
- Derivation for Call Prem: (and put prem is similar)
- Call Prem = \(e^{-rt}E[Call ~ Payoff]\) = \(e^{-rt}[S_0e^{(r-\delta)t}N(d_1) - KN(d_2)]\) = \(S_0e^{-\delta t}N(d_1) - Ke^{-rt}N(d_2)\)
- When dealing w/ discrete dividends we can use prepaid fwd equations (these formulas also work w/ continuous divs)
- \(C = F^p(S)N(d_1) - F^p(K)N(d_2)\)
- \(P = F^p(K)N(-d_2) - F^p(S)N(-d_1)\)
- \(d_1 = \frac{log(\frac{F^p(S)}{F^p(K)}) + .5 \sigma^2 t}{\sigma \sqrt{t}}\)
- \(d_2 = d_1 - \sigma \sqrt{t}\)
- \(F^p(S) = S_0 - \sum div_i e^{-rt_i} = S_0 e^{-\delta t}\)
- Subtract the PV of each dividend from the stock price, or discount by \(\delta\)
- \(F^p(K) = Se^{-rt}\)
- Just discount \(S_0\) by the rfr
True Pricing
Use same equations as the above, but substitute \(\alpha\) for \(r\), and instead of discounting by \(r\), we discount by \(\gamma\)
- Call Prem = \(e^{-\gamma t}[S_0e^{(\gamma-\delta)t}N(d_1)-KN(d_2)]\)
- To avoid having to memorize this new equation, just memorize the regular equations above, and first compound the equation by r, then substitute \(\gamma\) for \(r\)
The GREEKS
| GREEK | Formula from IFM Tables | Explanation | Values/Magnitude |
|---|---|---|---|
| \(\Delta\) | Call = \(e^{-\delta(T-t)}N(d_1)\) Put = \(-e^{-\delta(T-t)}N(-d_1)\) |
Change in Option price as stock increases by \(1|Higher Magnitude when the profit is higher. + for Calls, - for Puts| |\)$ | Call & Put = \(\frac{e^{-\delta(T-t)}N'(d_1)}{S\delta\sqrt{T-t}}\) |
| Vega or \(\kappa\) | Call & Put = \(Se^{-\delta(T-t)}N'(d_1)\sqrt{T-t}\) | Change in option price when volatility increases by 1% | High when stock and strike prices are close. Usually + |
| \(\theta\) | big equation | Change in option price when time moves forward one day closer to maturity | Usually - b/c options usually lose value w/ smaller time frame, but can be + if deep in the money |
| \(\rho\) | big equation | Change in option price when interest rate increases by 1% | Higher mag when in the money & T is longer. Calls = +, Puts - |
| \(\psi\) | big equation | Change in option price when dividend yield \(\delta\) increases by 1% | Higher mag when in the money & T is longer. Calls = -, Puts = + |
- Don’t need to memorize the equations, but the need to memorize/understand the last 2 columns
Greeks of multiple positions
\(\Delta = \sum_{i=1}^{N} (n_i)(greek_i)\)
- For a portfolio of options, you just sum up all the \(\Delta\)’s or \(\Gamma\)’s or any Greek of each option, to get the net Greek value.
- When long, \(\Delta\) => 1. When Short => \(\Delta\) => -1.
Other Greek-ish formulas to memorize
Option Elasticity (\(\Omega\))
- \(\Omega = \frac{S\Delta}{V}\)
- V = Option premium for Call or Put
- \(\Omega\) is the 3rd derivative of the options price, or derivative of \(\Gamma\)
- Represents: If the stock changes by 1%, by what % does the option change?
- \(\Omega = \frac{S\Delta}{V}\)
Option Volatility (\(\sigma_{option}\))
- \(\sigma_{option} = \sigma_{stock} |\Omega|\)
- Option volatility is higher than stock volatility
- \(\sigma_{option} = \sigma_{stock} |\Omega|\)
Risk Premium
- Risk Premium on a Stock = \(\alpha - r\)
- \(\alpha\) = Expected rate of return on a stock
- \(r\) = rfr (t-bill)
- Risk Premium of an Option = \(\gamma - r = (\alpha - r)\Omega\)
- \(\gamma = \Omega\alpha + (1-\Omega)r\) = Expected return of an option
- Risk Premium on a Stock = \(\alpha - r\)
Sharpe Ratio
- Sharpe Ratio \(=\frac{\alpha - r}{\sigma}\)
- Ratio between risk premium and volatility
- Same equation for Stocks and Options (but for Options I think we use \(\gamma - r\) for the numerator b/c risk premium)
- We want this value to be high b/c we want high return for low risk
Delta Hedging
About
- Delta hedging might be the only thing actuaries actually use from IFM material
- Delta hedging is offsetting an options position by buying/selling \(\delta\) shares of stock
- Market Maker is the person who sells the options contract
Delta Gamma Theta (\(\Delta\Gamma\theta\)) approximations
- \(C_{t+h}(S+\epsilon)\approx C_t(S) + \Delta\epsilon + .5\Gamma\epsilon^2 + \theta h\)
- A way to approximate the change in options price given the change in a stock price
- Only use part of the formula for however much they want you to approximate
Marking to Market
- Market maker adjusts his portfolio every day to stay hedged against market swings
- Best demonstrated with an example: Customer buys a t=91/365 day call option, so market maker sells the call option. Suppose S = $40, K = $40, \(\sigma\) = .3, \(r\) = .08, \(\delta\) = 0, and call contract is for 100 shares.
- Using Black Scholes framework and Greek Formulas we obtain these values in respect to the Call buyer:
- C(40) = 2.7804, \(\Delta\) = .5824, \(\Gamma\) = .0652, \(\theta\) = -.0173
- For the market maker, multiply each of these by -1 b/c we are selling the call
- C(40) = -2.7804, \(\Delta\) = -.5824, \(\Gamma\) = -.0652, \(\theta\) = .0173
- To hedge, the market maker buys .5824 shares of stock for every call
- Day 0:
- Market maker sells option contract so receives premium
- 2.7804*100 = 278.04
- Market maker buys \(\Delta\) shares for every option to hedge and spends:
- -.5824(100)(40) = -2329.6
- Market maker net position at t=0
- 278.04 - 2329.6 = -2051.56
- B/c it’s negative, we must borrow this money and will pay interest on it. If it were +, we would be lending this money and gaining interest
- Market maker sells option contract so receives premium
- Day 1: Stock rises to $40.50. Using Black Scholes w/ t=90/365, C(40) => $3.0621
- Market maker profit on shares he owns
- (40.50 - 40)(58.24) = 29.12
- Market maker profit on change in option value
- (2.7804 - 3.0621)(100) = -28.17
- Interest expense market maker must pay. We take previous period balance and compound by rfr
- \(-2051.56(e^{.08(\frac{1}{365})}-1) = -0.45\)
- Market Maker’s overnight profit:
- \(29.12 + -28.17 + -0.45 = 0.5\)
- Market maker must rebalance his portfolio
- Use Black Scholes to calculate new \(\Delta\) to be => .6142. So we must buy additional shares:
- (.6142 - .5824)(100) = 3.18
- So we must buy 3.18 more shares at new $40.50 price to rebalance portfolio: = 3.18(40.5) = 128.79
- This isn’t used in the profit equation for the day though
- Use Black Scholes to calculate new \(\Delta\) to be => .6142. So we must buy additional shares:
- Market maker’s new net position at Day 1:
- (3.0621)(100 options we sold) - (40.5)(61.42 shares we paid for) = -2181.3
- Market maker profit on shares he owns
- Day 2: Stock falls to $39.25, t=89/365, using Black Scholes C(40) => 2.3282
- Market maker profit on 61.42 shares he now owns
- (39.25 - 40.5)(61.42) = -76.775
- Market maker profit on option value change
- (3.0621 - 2.3282)(100) = 73.39
- Market maker pays interest on previous loan balance
- \(-2181.3(e^{.08/365}-1) = -0.48\)
- Market maker’s overnight profit:
- \(-76.775 + 73.39 + -0.48 = -3.865\)
- Market maker profit on 61.42 shares he now owns
- Here’s a summary table of what happens each day:
- Using Black Scholes framework and Greek Formulas we obtain these values in respect to the Call buyer:
- Steps in Marking to Market
- Find market maker’s net Delta position, and offset by buying/selling that many shares
- Find market maker’s net position from shares and option premiums
- For the next day
- Use new Stock price and t in the Black Scholes framework to find new Delta and Call premium
- Find the profits for your stock and option premium positions
- Find profit for how much you pay/receive in interest from previous position
- Add/subtract shares from portfolio to mark to market (rebalance portfolio) w/ new Delta
- Find new Net position
- Remember, interest profit never gets added to the net portfolio value
Gamma Hedging
- To stay even more hedged, not only do we make our net \(\Delta\) of our position = 0, but we also make our net \(\Gamma\) of our position = 0.
- We do this by buying/selling shares of another call option to get a net \(\Gamma\) of 0, but then have to buy/sell stock shares to get a net \(\Delta\) of 0
- We have 2 options w/ strikes: \(K_1\) and \(K_2\), each w/ their own \(\Delta\)’s and \(\Gamma\)’s
- If we sell the \(C(K_1)\) option, we \(\Gamma\) hedge by buying \(\frac{\Gamma_1}{\Gamma_2}\) \(K_2\) Call options to make our net \(\Gamma\) 0
- Our combined \(\Delta\) becomes:
- \(\Delta_{combined} = -\Delta_1+\frac{\Gamma_1}{\Gamma_2}\Delta_2\)
- You can logic your way to this equation if you just remember to buy \(\frac{\Gamma_1}{\Gamma_2}\) contracts of option 2
- So if this is negative we purchase that many shares to offset our position
Exotic Options
Asian Options
- Deals with the average price of the stock
- 2 different kinds of averages
- Arithmetic Average A(T) = just regular mean
- Geometric Mean = \((S_1*S_2*S_3*S_4*...)^{1/N}\)
- Recall that payoffs are calculated:
- Call payoff = Max{0, S-K}
- Put payoff = Max{0, K-S}
- For Asian options, we replace either S, or K using either A(T) or G(T)
- Between Call/Put, S/K, A(T)/G(T) there are 8 diff combinations
- Prob will say “avg price” for replacing S, or “avg strike” for replacing K
Barrier Options
- Knock-in Option (“Up and In”) = Can only exercise option if at any point between time 0 and T, the stock price crosses the barrier at least once
- Knock-out Option (“Down and Out”) = Option goes out of existence if at any point between time 0 and T, the stock price crosses the barrier once.
- These are priced using computer simulations
- Parity relationship for option premiums
- Knock-in + Knock-out = Ordinary option
Compound Options
- Option to buy an Option
- Have 2 strikes and 2 expirations
- \(t_0\) is current time. At \(t_1\), you have the option to buy (for a price of \(\$x\)) a European option with a strike price of \(\$K\). At time \(T\), the underlying European option expires.
- \(S^*\) is the critical value that if the stock rises above this value, then it makes sense to exercise the compound option.
Compound Option Parity
- CallonCall - PutonCall = Call - \(xe^{-rt_1}\)
- CallonPut - PutonPut = Put - \(xe^{-rt_1}\)
Gap Options
- Gap options have a regular strike price \(K_1\) that you buy/sell the asset for, but you can only do so if the stock is more/less than the “trigger” price \(K_2\) at expiration
- Use the same Black Scholes formulas except we change the \(d_1\) formula to have \(K_2\) in place of \(K_1\)
- \(C = Se^{-\delta t}N(d_1)-K_1e^{-rt}N(d_2)\)
- \(P = K_1e^{-rt}N(-d_2) - Se^{-\delta t}N(-d_1)\)
- \(d_1 = \frac{log(\frac{Se^{-\delta t}}{K_2e^{-rt}}) + .5\sigma^2t}{\sigma\sqrt{t}}\)
- \(d_2 = d_1 - \sigma\sqrt{t}\)
Exchange Options
- AKA Outperformance Options
- Basically an option to exchange an asset with another
- S is the price of risky asset 1, K is the price of risky asset 2
- We are given dividend yields (\(\delta_S\) & \(\delta_K\)), volatilites (\(\sigma_S\) & \(\sigma_K\)), and correlation between the 2 continuously compounded dividend yields (\(\rho\))
- Call option is giving asset K (2), and receiving asset S (1). (Paying strike K, receiving stock S) (Same as normal)
- Call Prem = \(Se^{-\delta t}N(d_1) - Ke^{-rt}N(d_2)\)
- \(d_1 = \frac{log(\frac{Se^{-\delta_S t}}{Ke^{-\delta_K t}}) + .5\sigma^2t}{\sigma\sqrt{t}}\)
- \(d_2 = d_1 - \sigma\sqrt{t}\)
- \(\sigma = \sqrt{\sigma^2_S + \sigma^2_K -2\rho\sigma_S \sigma_K}\)
Properties of Lognormal
Assume that a stock’s continuously compounded returns follow a normal distribution. And thus the stock price follows a lognormal distribution.
\(\frac{S_t}{S_0} \tilde{} LogN(m,v^2)\)
\(ln(S_t) \tilde{} N(m = ln(S_0)+(\alpha-\delta-.5\sigma^2)t, v^2 = \sigma^2 t)\)
- \(S_t = S_0 e^{(\alpha-\delta-.5\sigma^2)t + \sigma \sqrt{t} *Z}\)
- Where \(Z \tilde{} N(0,1)\)
- Coaching Actuaries says we should memorize this
Following Lognormality, we get these equations:
\(E[S_t]=S_0e^{(\alpha-\delta)t}\)
\(Med[S_t]=S_0e^{(\alpha-\delta-.5\sigma^2)t}=E[S_t]e^{-.5\sigma^2 t}\)
- \(E[S_t|S_t>K]=S_0e^{(\alpha-\delta)t}\frac{N(d_1)}{N(d_2)}\)
- \(E[S_t|S_t<K]=S_0e^{(\alpha-\delta)t}\frac{N(-d_1)}{N(-d_2)}\)
- \(d_1 = \frac{log(\frac{S_0}{K})+(r-\delta+.5\sigma^2)t}{\sigma\sqrt{t}}\)
- Assuming no arbitrage \(\alpha = r\)
- \(d_2 = d_1 - \sigma\sqrt{t} = \frac{log(\frac{S_0}{K})+(r-\delta-.5\sigma^2)t}{\sigma\sqrt{t}}\)
- Only diff is we subtract \(.5\sigma^2\) from the numerator
- \(d_1 = \frac{log(\frac{S_0}{K})+(r-\delta+.5\sigma^2)t}{\sigma\sqrt{t}}\)
- \(P(S<K)=N(-d_2)\)
- \(P(S>K)=N(d_2)\)
- \(d_1 = \frac{log(\frac{S_0}{K})+(\alpha-\delta+.5\sigma^2)t}{\sigma\sqrt{t}}\)
- Only diff is we use \(\alpha\) instead of \(r\) here b/c we assume no arbitrage
- \(d_2=d_1-\sigma\sqrt{t}\)
- \(d_1 = \frac{log(\frac{S_0}{K})+(\alpha-\delta+.5\sigma^2)t}{\sigma\sqrt{t}}\)
- \(Var(S_t) = E(S_t)^2(e^{\sigma^2 t}-1)\)
- Pearson Textbook way: (harder)
- \(Var(S_t)=S_0^2 Var(X)\)
- \(Var(X)=e^{2m+v^2}(e^{v^2}-1)\)
- \(m=(\alpha-\delta-.5\sigma^2)t\)
- \(v^2=\sigma^2 t\)
- \(Var(X)=e^{2m+v^2}(e^{v^2}-1)\)
- \(Var(S_t)=S_0^2 e^{2t(\alpha - \delta -.5\sigma^2)+\sigma^2 t}(e^{\sigma^2 t}-1)\)
- \(Var(S_t)=S_0^2 Var(X)\)
Variance/Covariance/Correlation Review
- \(Var(X) = E[(X-E(X))^2] = E(X^2)-E(X)^2\)
- \(S^2 = \frac{\sum (x_i -\bar{x})^2}{N-1}\)
- Unbiased estimator of the variance. Use when dealing w/ a sample
- \(\sigma^2 = \frac{\sum (x_i -\bar{x})^2}{N}\)
- Actual variance. Use when the population mean is known
- \(Var(aX+bY) = a^2Var(X)+b^2Var(Y)+2abCov(X,Y)\)
- Cov = 0, when X and Y are independent
- Cov is usually given on the exam
- \(Var(X+Y+Z) = Var(X)+Var(Y)+Var(Z)+Cov(X,Y)+Cov(X,Z)+Cov(Y,Z)\)
- Variance of each, plus covariance of every combination
- \(Cov(X,Y)=\frac{\sum (x_i -\bar{x})(y_i -\bar{y})}{N-1}\)
- This is the unbiased estimator, use only N in the denominator for actual covariance
- \(Cov(X,Y) = E(XY)-E(X)E(Y)\)
- \(\rho=Corr(X,Y)=\frac{Cov(X,Y)}{SD(X)SD(Y)}\)
Corporate Finance
Chap 10
Historical Return
- \(R_{t+1}=\frac{Div_{t+1}+P_{t+1}}{P_t}-1\)
- \(R\): Return
- \(Div_t\): dividend amount received at t=t
- \(P_{t}\) Price at t=t
- Basically this is just how much you have at the end, divided by how much you started w/, -1 to get the return.
Quarterly -> Annual returns
- \(1+R_{annual}=(1+R_{Q1})(1+R_{Q2})(1+R_{Q3})(1+R_{Q4})\)
- These can be any time interval, as long as they all add up to 1 year. Can even combine weeks and months
Risk
- Common Risk: Systematic Risk, Undiversifiable Risk, Market Risk
- Ex: Terrorist attack
- Independent Risk: Firm specific risk, idiosyncratic risk, unique risk, unsystematic risk, diversifiable risk
- Ex: scandal at 1 firm
Sensitivity to systematic risk (\(\beta\))
- Expected % change of the excess return of a security, for a 1% change in the excess return of a mkt portfolio.
- \(\beta = \frac{Range of returns for a stock}{Range of returns of market}\)
- Or calculate \(\beta\) through regression techniques
CAPM Capital Asset Pricing Model
- \(E[R]=r_f +\beta(E[R_{mkt}]-r_f)\)
- Used to find expected return on a stock
Chapter 11
Return on Portfolio
- Actual return in portfolio
- \(R_p=\sum x_i R_i\)
- \(x_i\) = portfolio weight of asset i
- \(R_i\) = return from asset i
- \(R_p=\sum x_i R_i\)
- Expected return in portfolio
- \(E[R_p]=\sum x_i E[R_i]\)
Volatility of portfolio
- Main Equation for the variance of a portfolio of returns
- \(Var(R_p)=\sum \sum x_i x_j Cov(R_i, R_j)\)
- You do this for every combination of returns, w/ combinatorics factors
- For ex, this is for a portfolio of 3 stocks
- \(Var = 2x_1x_2Cov(R_1, R_2) + 2x_1x_3Cov(R_1, R_3) + 2x_2x_3Cov(R_2, R_3) +\) \(x_1^2Cov(R_1, R_1) + x_2^2Cov(R_2, R_2) + x_3^2Cov(R_3, R_3)\)
- And recall that the Cov() of a return w/ itself is just the variance
- \(Var = 2x_1x_2Cov(R_1, R_2) + 2x_1x_3Cov(R_1, R_3) + 2x_2x_3Cov(R_2, R_3) +\) \(x_1^2Cov(R_1, R_1) + x_2^2Cov(R_2, R_2) + x_3^2Cov(R_3, R_3)\)
- \(Var(R_p)=\sum \sum x_i x_j Cov(R_i, R_j)\)
- Covariance and Correlation formulae
- \(Cov(R_i, R_j)=E[(R_i-E[R_i])(R_j-E[R_j])] = \frac{1}{T-1} \sum (R_{i,t} - \bar{R_i})(R_{j,t} - \bar{R_j})\)
- \(Corr(R_i, R_j)=\frac{Cov(R_i, R_j)}{SD(R_i)SD(R_j)}\)
Efficient Portfolio
- A portfolio is efficient if you have the lowest amount of volatility for a given return
Adding T Bills
- Do this to reduce some risk
- Expected return with adding T bills to portfolio
- \(E[R_{xp}] = xE[R_p] + (1-x)r_f\)
- x = %age of portfolio that consissts of stocks
- \(r_f\) = rfr from investing in bonds
- \(E[R_{xp}] = xE[R_p] + (1-x)r_f\)
- Standard Deviation
- \(SD(R_{xp}) = xSD(R_p)\)
- Standard deviation of a portfolio with t bills, is just the %age of stocks in portfolio, times the SD of the stock portfolio. B/c there is no standard deviation w/ T bills.
- \(SD(R_{xp}) = xSD(R_p)\)
Portfolio improvement
- \(\beta\) of 2 diff assets, one w/ respect to the other. How much the movement of 1 can be described by the movement of another
- \(\beta_i^P = \frac{SD(R_i)Corr(R_i, R_P)}{SD(R_P)}\)
- Beta of stock i, w/ portfolio P (market portfolio)
- Beta of a portfolio (just weighted avg of all the \(\beta\)’s)
- \(\beta_P = \sum x_i \beta_i\)
Required Rate of return
- \(r_{rrr}=r_f+\beta_i^P(E[R_P]-r_f)\)
- Subjective, diff for each person. What they want to receive in return
- This should be what you require to return for adding a new stock, i, to your portfolio. Calculate the required return using the above, then if the individual stock gives you a higher return than the required rate of return, then it makes sense to add the stock to your portfolio.
Chap 12 Cost of Capital
Equity Cost of Capital
- Use CAPM to calculate cost of capital. (Return on equity)
- \(r_E=E(R_i)=r_i + \beta(E[R_{mkt}]-r_i)\)
Debt Cost of Capital
- Method 1: Use expected return of a bond
- \(r_d=y-PL\)
- \(r_d\) = return on debt
- \(y\) = yield to maturity of bond
- \(P\) = probability of default
- \(L\) = Expected loss rate. Expected loss per $1 of debt.
- \(r_d=y-PL\)
- Method 2: Use market risk premium (CAPM style)
- \(r_d = r_f + \beta(E[R_{mkt}]-r_f)\)
- \(\beta\) = The \(\beta\) used here is diff than our equity \(\beta\). Risker bonds have higher \(\beta\), so we look it up on a table based on it’s rating (AAA, A, BBB, BB, …)
- \(r_d = r_f + \beta(E[R_{mkt}]-r_f)\)
Asset (unlevered) Cost of Capital
- Expected return required by investors to hold the firm’s underlying assets
- Weighted avg of the firm’s cost of equity and cost of debt
- \(r_U = \frac{E}{E+D}r_E + \frac{D}{E+D}r_D\)
- \(r_U\) = unlevered return
- \(r_E\) = return on equity
- \(r_D\) = return on debt
- \(E\) = Amt of equity
- \(D\) = Net Debt = Debt - Cash
- \(r_U = \frac{E}{E+D}r_E + \frac{D}{E+D}r_D\)
- Asset (Unlevered) Beta
- \(\beta_U = \frac{E}{E+D}\beta_E + \frac{D}{E+D}\beta_D\)
- This is the underlying beta of the business enterprise
- \(\beta_U = \frac{E}{E+D}\beta_E + \frac{D}{E+D}\beta_D\)
- Random formula
- Enterprise Value = Mkt Cap + Net Debt
- Mkt Cap is equity
- Enterprise Value = Mkt Cap + Net Debt